If ##r \notin S\cup \{\,0\,\}##, then the minimality of ##d## has nothing to do with ##r## and we cannot conclude ##r=0##. It is zero, because it is part of this set!Suyogya said:what is the need to show that r belongs to S U {0} in proof (https://en.wikipedia.org/wiki/Bézout's_identity#Proof)
r is zero afterall, whether it lies in S U {0} or not doesn't affect.
please tell the following:fresh_42 said:If ##r \notin S\cup \{\,0\,\}##, then the minimality of ##d## has nothing to do with ##r## and we cannot conclude ##r=0##. It is zero, because it is part of this set!
We start with a set ##M##. Then we choose a minimal element of ##M##, called ##m##. In order to compare any other element ##n## to ##m##, we can only do this, if ##n \in M##, because then we know, that ##m \leq n## as ##m## was chosen minimal. Otherwise we can't say anything.Suyogya said:please tell the following:
is the need to show that r lies in S U {0} just because to compare r is less than or equal to d (as d also belongs to S U {0})?
So are only the same set elements can be compared? If yes, then consider a counter example, set a={10}, set b={5} couldn't we say 5<10 (as both belongs to different sets).
The Bezout Identity is a fundamental theorem in algebra that states that for any two nonzero integers, there exist integers x and y such that ax + by = gcd(a,b), where gcd(a,b) is the greatest common divisor of a and b.
The Bezout Identity is used in many areas of mathematics, including number theory, algebraic geometry, and abstract algebra. It is often used to prove the existence of solutions to Diophantine equations and to find the inverse of an element in a ring.
The condition r∈S U {0} is necessary in proving the Bezout Identity because it ensures that the greatest common divisor of a and b is a nonzero integer. Without this condition, the theorem would not hold true.
Yes, the Bezout Identity can be extended to other number systems, such as polynomials, matrices, and even complex numbers. However, the proof and conditions may vary for each system.
The Bezout Identity has many practical applications, including cryptography, error-correcting codes, and signal processing. It is also used in computer graphics to determine the intersection of two curves or surfaces.